NDLI: On convergent numerical algorithms for unsymmetric collocation

Content Provider	Springer Nature Link
Author	Schaback, Robert Lee, Cheng Feng Ling, Leevan
Copyright Year	2008
Abstract	In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.
Ending Page	354
Page Count	16
Starting Page	339
File Format	PDF
ISSN	10197168
e-ISSN	15729044
Journal	Advances in Computational Mathematics
Issue Number	4
Volume Number	30
Language	English
Publisher	Springer US
Publisher Date	2008-05-07
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Numeric Computing Kansa’s method Effective condition number Theory of Computation Mathematics Convergence Radial basis function Stability and convergence of numerical methods Algebra Spectral, collocation and related methods High precision computation Calculus of Variations and Optimal Control; Optimization Error bounds Linear optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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